_______________________________
A Benders Decomposition Approach for the Symmetric TSP with
Generalized Latency
Fausto Errico
Teodor Gabriel Crainic Federico Malucelli Maddalena Nonato
December 2012
CIRRELT-2012-78
G1V 0A6 Bureaux de Montréal : Bureaux de Québec :
Université de Montréal Université Laval
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www.cirrelt.ca
Fausto Errico
1,2, Teodor Gabriel Crainic
1,3,*, Federico Malucelli
4, Maddalena Nonato
51 Interuniversity Research Centre on Enterprise Networks, Logistics and Transportation (CIRRELT)
2 Department of Mathematics and Industrial Engineering, École Polytechnique de Montréal, P.O.
Box 6079, Station Centre-ville, Montréal, Canada H3C 3A7
3 Department of Management and Technology, Université du Québec à Montréal, P.O. Box 8888, Station Centre-Ville, Montréal, Canada H3C 3P8
4 Dipartimento di Elettronica ed Informazione, Politechnico di Milano, Piazza L. Da Vinci 32, 20133 Milano, Italy
5 Dipartimento di Ingegneria, Università degli studi di Ferrara, Via Saragat 1, 44100, Ferrara, Italy
Abstract. We present a new problem within the class of TSP. The problem arises in the transportation field, namely in the design of semi-flexible transit lines, but it may have applications also in other areas. The objective function combines the usual cost of the Hamiltonian circuit with a routing cost that depends on the actual flow on each arc. The main contributions include the formulation in terms of multicommodity flow, the study of mathematical properties and a Branch-and-Cut approach based on Benders decomposition. Some variants of the algorithm are experimentally evaluated, in comparison with a commercial solver. Several algorithmic refinements are proposed and their efficacy is experimentally tested on a variety of instances. Tests on benchmark instances confirm the validity of our approach.
Keywords : Traveling salesperson problem, latency, Benders decomposition, branch-and- cut, semi-flexible transit.
Acknowledgements. While working on this paper, F. Errico was postdoctoral researcher at École des sciences de la gestion, Université du Québec à Montréal. T.G. Crainic was the NSERC Industrial Research Chair in Logistics Management, ESG UQAM, and Adjunct Professor with the Department of Computer Science and Operations Research, Université de Montréal, and the Department of Economics and Business Administration, Molde University College, Norway, and F. Errico was postdoctoral researcher with the Chair.
Funding for this project was provided by the Natural Sciences and Engineering Research Council of Canada (NSERC), through its Industrial Research Chair and Discovery Grant programs, by our partners CN, Rona, Alimentation Couche-Tard, the ministère des Transports du Québec, and by the Fonds de recherche du Québec - Nature et technologies (FRQNT) through its Team Research Project program. We also gratefully acknowledge the support of the Canada Foundation for Innovation, the Québec Ministry of Education, UQAM and SUN/Azuris for our computing infrastructure, as well as that of Fonds de recherche du Québec for the CIRRELT infrastructure grants.
Results and views expressed in this publication are the sole responsibility of the authors and do not necessarily reflect those of CIRRELT.
Les résultats et opinions contenus dans cette publication ne reflètent pas nécessairement la position du CIRRELT et n'engagent pas sa responsabilité.
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* Corresponding author: [email protected] Dépôt légal – Bibliothèque et Archives nationales du Québec Bibliothèque et Archives Canada, 2012
1 Introduction
In this paper, we introduce a new optimization problem, the TSP-GL, based on the Symmetric Traveling Salesperson Problem (S-TSP) to which we add a generalized latency component in the objective function.
This problem has immediate application in the network design of transit lines but it may also be applied in other fields.
The problem can be described as follows. As in the S-TSP, we are given a complete undirected graph with non negative costs on the edges and a Hamiltonian circuit has to be found. In addition, we are given an origin/destination demand matrix, where each entry specifies the amount of demand to be routed between the corresponding origin and destination nodes of the graph. Moreover, we are given the traversing times or unit routing costs for every edge and for both directions. Therefore, besides finding a Hamiltonian circuit, we must also find the routing of the demand on the circuit. The aspect that characterizes theTSP-GLand makes the problem very challenging is the objective function that accounts for two components. The first one considers the cost of the edges in the Hamiltonian circuit (installation cost), the second accounts for the overall time spent to traverse the required portion of the circuit by each demand unit (routing cost or latency).
In the design of public transit services, passenger travel times constitute a key factor directly affecting the level of service. Traditionally, passenger travel times are accounted for at tactical planning level, i.e., after the line design has been established. Passenger travel times optimization is only considered when defining service frequencies and timetables (see Desaulniers and Hickman 2007, for an extensive review on public transit planning). In such a context, theTSP-GL plays an important role, allowing for the simultaneous optimization of passenger travel times (latency) and operating costs (installation).
For illustration sake, let us briefly describe a basic version of the design of a single circular transit line, operated in both directions, in terms of TSP-GL optimizing both operational costs and passenger travel times. The nodes of the graph correspond to the bus stops, the edges are the possible direct connections between two stops. The installation costs and the routing costs are given by the traveling cost and shortest traveling time between the corresponding pair of bus stops, respectively. The origin/destination matrix is the estimated transportation demand between bus stop pairs. In planning the line we must design a circuit passing by all the bus stops exactly once. The objective is to minimize a function that combines the traveling cost (operational cost) and the overall time spent by users on vehicles (level of service). Given a circuit, for any origin there are two possible alternative itineraries to reach the destination, depending on the direction taken on the circuit. We assume that users choose the shortest one in terms of traveling time. The choice of considering a circular line does not limit the generality of the approach, that can be easily adapted to Hamiltonian paths.
TheTSP-GLhas been applied to the design of a wide class of on-demand transit services called semi- flexible. Errico et al. (2011b), presents an extensive computational study of the application of TSP-GLto the design of semi-flexible transit lines, evaluating the influence of the two components on the final results.
For an exhaustive review on the state-of-the-art and state-of-the-practice of semi-flexible transit systems see Errico et al. (2011a) and Potts et al. (2010).
The TSP-GL is at least as difficult as the TSP. In fact, if the latency component is removed from the objective function, the TSP-GLreduces to a S-TSP, which immediately proves the complexity of the problem. However, theTSP-GLis much harder than the TSP due to the particular objective function. Note that, given a Hamiltonian circuit, determining the routing of each demand pair is straightforward since we must only select in which direction it has to be sent along the circuit, and this can be done separately and independently for each demand pair. In spite of this fact, the nature of the objective function makes the
problem decisions extremely correlated and computationally very challenging, as we will see further on.
At the best of our knowledge, we are the first to introduce and formalize the TSP-GL. However, there are two families of problems known in the literature which are related to theTSP-GL: The Fixed-Charge Network Design Problem (FNDP) and several variants of the TSP. Nonetheless, the specific structure of the TSP-GLand its complexity do not allow for a successful adaptation of existing approaches to the above mentioned problems and actually, new ad hoc solution methods are required in order to solve practical size instances.
We propose a multicommodity flow based formulation for theTSP-GLas an extension of the traditional S-TSP formulation where the new variables are used to express our specific objective function. Such a formulation has the advantage of being compact, having a polynomial number of variables and constraints with respect to the dimension of the problem. However, this formulation presents some drawbacks: In spite of its compactness, the solution of the LP relaxation requires very long computing times, and the lower bounds obtained this way are quite loose. These reasons gave us the motivation to investigate a solution approach that combines Benders decomposition and generation of valid inequalities. The Benders decomposition, in fact, requires to iteratively solve a master problem, which is a relaxation of the original problem, and a (dual)subproblem used to generate inequalities improving the current solution of the master problem. The inequalities generated are of two kinds: optimality cuts andfeasibility cuts.
Differently from the traditional scheme where Benders cuts are only generated at integer master solutions, in this paper we consider an alternative scheme where integrality constraints of the master problem are relaxed, and integrality is recovered by embedding the Benders master problem into a genuine Branch and Cut framework. We call such an approach the Benders-Branch-and-Cut algorithm (BBC). The BBC algorithm seems particularly attractive for theTSP-GLbecause, by projecting flow variables and relaxing integrality, the similarity betweenTSP-GLand S-TSP can be effectively exploited.
Starting from the traditional Benders reformulation, we consider three alternative families of inequalities to be deployed as feasibility cuts. We prove mathematical relations among the lower bounds provided by each family. We implemented the corresponding versions of the BBC algorithm and experimentally com- pared their efficiency. All BBC algorithms widely outperform a state-of-the-art MIP solver applied to the initial multicommodity flow formulation. As a second step, we refined the most efficient BBC algorithm by implementing some techniques known to improve the performance of the traditional Benders decomposition.
We experimentally compare such refinements and the resulting best choices are used to attack a set of newly created benchmark instances derived from the well-known TSPLIB library. It turns out that the refined BBC algorithm is able to optimality solve instances with up to 42 nodes, while in the most difficult 101-node instance, it provides a feasible solution within 8.8% of the optimality in 5 hours of computing time.
The rest of the paper is organized as follows: In Section 2, we introduce theTSP-GLmulticommodity flow based formulation and we survey the related works. In Section 3, we recall the main ideas of the Benders decomposition and we apply Benders reformulation to the TSP-GL. Section 4 describes the general BBC and motivates our choice. In the same section, three versions of the BBC algorithm and their mathematical properties are examined. In Section 5, we report the details of the experimental campaign, computational results and analyses. We draw our conclusions in Section 6.
2 Problem statement and related works
The TSP-GL problem involves two levels of decisions. On the one hand the set of edges defining the Hamiltonian circuit must be selected, on the other hand the demand has to be routed on the circuit and the routing direction is an issue. This suggests the employment of a mixed graph since both edges and directed
arcs are used.
Consider a complete mixed graphG= (N, E∪A), whereN ={1, . . . , n} is the set of nodes,E={[i, j] : i, j∈N, i < j} is the set of edges, andA ={(i, j) : i, j ∈ N, i6=j} is the set of directed arcs. Adesign cost ¯ce > 0 is associated to each edge e = [i, j] ∈ E. Moreover, the amount of demand dhk is given for each origin and destination node pair (h, k). Let D denote the set of pairs having non zero demand, that isD ={(h, k) : h, k ∈N, dhk >0}. A routing costqhkij ≥0 is associated to every arc (i, j)∈A and every demand (h, k)∈D.
The problem consists in finding a subsetX ⊂E of edges defining a Hamiltonian circuit inGand a flow such that the demand is satisfied, that is the flow going from hto k is equal to dhk for each pair of nodes (h, k) ∈ D and the flow is consistent with the circuit definition, that is the flow on arc (i, j) for any pair (h, k)∈D can be different from 0 only if edgee= [i, j] belongs to the circuitX. The objective function to be minimized is the sum of the design costs of the edges inX and of the routing costs of the arcs traversed by positive flow.
2.1 Multicommodity flow based formulation
The most natural way to formulate the problem is to use an integer multicommodity flow based model. For each edgee= [i, j]∈E we introduce a binary variablexewith value 1 if the edge [i, j] is in the cycleX and 0 otherwise. We have then multicommodity flow variables for each commodity (h, k)∈D and for each arc (i, j)∈A: fijhk assumes value 1 if demand fromhtok travels on arc (i, j) and 0 otherwise.
In order to simplify the formulation, we will use the following notation: given a subset of nodes S⊂N, δ(S)⊂E denotes the set of edges having exactly one endpoint inSandδ(i) is used as a shorthand ofδ({i}).
Similarly, δ+(S) ⊂A (δ−(S)⊂ A) denotes the set of arcs having tail in S and head in N \S (head in S and tail inN \S, respectively). We useδ+(i) (δ−(i)) as a shorthand of δ+({i}) (δ−({i})). Given a subset E∗⊆E and a subset A∗⊆Awe writex(E∗) forP
e∈E∗xeandfhk(A∗) forP
(i,j)∈A∗fijhk, respectively.
TheTSP-GLis formulated as follows:
min (1−α)X
e∈E
¯
cexe+α X
(i,j)∈A
X
(h,k)∈D
qhkij fijhk (1)
x(δ(i)) = 2 ∀i∈N (2)
fhk(δ+(i))−fhk(δ−(i)) =
1 ifi=h,
−1 ifi=k, 0 i6=h, k
∀(h, k)∈D, ∀i∈N (3)
fijhk ≥0 ∀(i, j)∈A, ∀(h, k)∈D (4)
xe−fijhk ≥0 ∀e= [i, j]∈E,∀(h, k)∈D (5)
xe−fjihk ≥0 ∀e= [i, j]∈E,∀(h, k)∈D (6)
xe∈ {0,1} ∀e∈E (7)
The objective function (1) accounts for the two components: the sum of design costs and the latency, whereαis a suitable tradeoff parameter. Constraints (2) impose that the selected edges incident to each node must be exactly two, thus they must form a cycle cover ofG. Constraint (3) are the multicommodity flow balance constraints stating that one unit of flow must be sent fromhtokfor every (h, k)∈D. Constraints (5) and (6) enforce the coherence betweenxandf variables, that is, any commodity (h, k) cannot travel on an arcs (i, j) or (j, i) if the corresponding edgee= [i, j] is not in the cycle. Observe that constraints (5) and
(6) could be alternatively expressed in the apparently stronger form:
xe−fijhk−fjihk ≥0 ∀e= [i, j]∈E,∀(h, k)∈D. (8) There is no practical advantage in using this alternative expression however, because no optimal solution of the linear relaxation of (1)-(7) would havefijhk andfjihkstrictly positive at the same time. We keep consequently the original formulation.
Note that, in general, our formulation does not avoid the presence of subtours. Subtours are made infeasible by the combination of coherence constraints and flow balance constraints ifDis dense enough. The only case when the presence of subtours is possible is whenDdefines inGmore than one connected component.
In this case, in order to generate Hamiltonian circuits we must introduce explicit cutset inequalities. Usually, the demand matrixD in transit line design is dense enough to avoid this occurrence. In our investigation, we focus on the case of matricesD inducing a single connected component.
2.2 Related works
The TSP-GL can be seen as a specialized version of the FNDP consisting in finding a set of links (and capacities) on which flow demands can be routed, minimizing design and/or routing cost. In the case of the TSP-GL, links have infinite capacities and the network topology is constrained to be a Hamiltonian circuit.
The FNDP is a very general framework able to model a wide variety of problems and for which several exact methods have been proposed. To cite a few: Benders decomposition (see Costa (2005) for a complete review on the topic), Lagrangean based techniques (see for example Crainic et al. 2001; Holmberg and Hellstrand 1998; Holmberg and Yuan 2000; Kliewer and Timajev 2005; Sellmann et al. 2002) and polyhedral approaches (see for example Atamt¨urk 2002; Barahona 1996; Bienstock and G¨unl¨uk 1994; G¨unl¨uk 1999; Leung and Magnanti 1989; Ortega and A. 2003; Raack et al. 2011).
More specifically, the Uncapacitated FNDP (UFNDP) has been addressed in Magnanti et al. (1986) by Benders decomposition. The authors applied a preprocessing phase based on a dual-ascent procedure that allowed to eliminate design variables and showed that the selection of Benders cuts is a key element for the success of the approach. Holmberg and Hellstrand (1998) proposed a Lagrangean relaxation of the flow conservation constraints and embedded such a relaxation scheme into a Branch and Bound algorithm. Ortega and A. (2003) considered a particular case of the UFNDP where one origin only is allowed and proposed a Branch and Cut algorithm. The so-calleddicut inequalities showed to be very effective in their approach.
The TSP-GL also recalls a similar problem called Generalized Minimum Latency Problem introduced in Errico (2008). In the Generalized Minimum Latency Problem, we are given a complete directed graph G = (N, A) were directed arcs are associated with a design cost and a routing cost. As in TSP-GL, origin/destination demand coefficients dhk are given. The problem consists in determining a Hamiltonian directed cycle and in routing the demand on the cycle so that the sum of the design costs of the selected arcs and the routing costs multiplied by the flow is minimized. The main difference with respect toTSP-GLis that the cycle is directed, design costs are not necessarily symmetric, and demand routing must follow the direction of the arcs of the cycle.
Two important problems studied in the literature can be seen as special cases of the Generalized Minimum Latency Problem depending on how the demand is specified. One is the Minimum Latency Problem (MLP), also known in literature as the Traveling Deliveryman Problem or Traveling Repairman problem. A repairman has to visit a given set of customers, who issued a request before the repairman departure. Starting from a depot, the repairman has to visit all customers minimizing the total customer waiting time that depends on the distance traveled by the repairman before he/she serves them. The MLP can be modeled as a particular
case of Generalized Minimum Latency Problem where, supposing that the depot is in node 1, d1k = 1 for k= 2, . . . , nand 0 for all the other origin/destination pairs. Moreover the design costs are equal to zero for all arcs. The other problem is the Multicommodity TSP which is very similar to the MLP problem with the exception that the demand coefficients between node 1 and the others can assume any value greater than 0 rather than being unitary.
Lucena (1990) formulates the MLP using a time dependent TSP and obtains lower and upper bounds via dynamic programming and Lagrangean relaxation for instances with up to 30 nodes. More recently, M´endez-D´ıaz et al. (2008) proposed an evolution of the previous approach exploiting the polyhedral structure of the problem and increasing the size of the solved instances to 40 nodes. Different lower and upper bounds obtained using cumulative matroids and Lagrangean decomposition was proposed by Fischetti et al. (1993).
The Multicommodity TSP is studied in Sarubbi and Luna (2007) and Sarubbi et al. (2008) where a classical Benders decomposition and a Quadratic Assignment approach are proposed.
We observe that Benders decomposition has been adopted for several problems related to the TSP-GL and this motivated our study. However, as detailed in Section 4, the particular structure of the TSP-GL does not allow for a straightforward extension of the known methods. We will exploit such a structure to develop an efficient cutting plane algorithm based on Benders decomposition and polyhedral properties of theTSP-GL, and embedd it in an implicit enumeration scheme.
3 Benders decomposition and TSP-GL
The Benders decomposition was introduced in Benders (1962) as a method to solve mixed integer programs.
It applies in particular to structured problems where we can identify a subset of “complicating” variables for which, when fixed, the problem reduces to an easy one. In our case it is easy to notice that ifxvariables defining the Hamiltonian circuit are fixed, the resulting restricted problem reduces to a set of independent minimum cost flow problems without capacity constraints. Let us analyze the Benders reformulation for the TSP-GLand illustrate the classical Benders decomposition.
Let us consider the following problem depending onxvariables only:
min (1−α)X
e∈E
¯
cexe+αz(x) (9)
x(δ(i)) = 2 ∀i∈N (10)
xe∈ {0,1} ∀e∈E (11)
Function z(x) gives the contribution of the flow variables (i.e., the routing cost component) to the overall objective function and depends on the circuit definition. Given a possible solution ¯x, the value of z(¯x) is obtained by solving the so calledP rimalSubproblem (PS). For the TSP-GL, since there are no capacities on the arcs, the PS may be decomposed into independent one-commodity minimum cost flows, one for each origin-destination pair: z(¯x) =P
(h,k)∈Dzhk(¯x), where zhk(¯x) = min X
(i,j)∈A
qhkij fijhk (12)
fhk(δ+(i))−fhk(δ+(i)) =
1 i=h,
−1 i=k,
0 ∀i∈N\ {h, k}
∀i∈N (13)
−fijhk≥ −¯xe ∀i, j∈N, i < j, e= [i, j] (14)
−fjihk≥ −¯xe ∀i, j∈N, i < j, e= [i, j] (15)
fijhk≥0 ∀(i, j)∈A. (16)
Observe that the strong duality holds for every flow subproblem, hence, we can equivalently consider the Dual Subproblems:
zhk(¯x) = max ρhkh −ρhkk −X
e∈E
λ¯hke x¯e (17)
ρhki −ρhkj −λhkij ≤qijhk, ∀(i, j)∈A (18)
λhkij ≥0 ∀(i, j)∈A (19)
ρhki ≷0 ∀(i, j)∈A (20)
where ¯λe:=λij+λji, ∀e= [i, j]∈E.
By denotingQ={(λ, ρ)∈ R(|A|+|N|)|D||(λ, ρ) satisfy (17), . . . ,(20)∀(h, k)∈D}, problem (9) - (11) can be rewritten as:
minx (1−α)X
e∈E
¯
cexe+α max
(λ,ρ)∈Q{ X
(h,k)∈D
(ρhkh −ρhkk −X
e∈E
λ¯hke xe)} (21)
x(δ(i)) = 2 ∀i∈N (22)
xe∈ {0,1} ∀e∈E (23)
The fact that flow variablesf do not explicitly appear in problem in (9) - (11), and as a consequence also in problem (21) - (23), implies that coherence constraints linkingxand f can be violated. Solutionsxthat are feasible for (9) - (11) might therefore not guarantee the feasibility of the PS, and DS would result to be unbounded. To avoid this circumstance, we impose in problem (9) - (11) (and equivalently in problem (21) - (23)) thatP
(h,k)∈D(ρhkh −ρhkk −P
e∈Eλ¯hke xe)≤0 for every (λ, ρ) extreme ray ofQ.
Since unbounded solutions are now avoided, the new DS has optimum solution in one of the extreme points ofQ. In order to obtain the Benders reformulation, we linearize the objective function (21). Because the number of extreme points of a polyhedron is finite, the expression (21) can be seen as the minimum of a set of affine functions which could be linearized with the addition of a continuous variableη.
Now we can write the Benders reformulation of theTSP-GL: min α η+ (1−α)X
e∈E
¯
cexe (24)
η+X
e∈E
X
(h,k)∈D
λ¯hke xe≥ X
(h,k)∈D
(ρhkh −ρhkk ) ∀(λ, ρ)∈extr(Q) (25) X
e∈E
X
(h,k)∈D
λ¯hke xe≥ X
(h,k)∈D
(ρhkh −ρhkk ) ∀(λ, ρ)∈rays(Q) (26)
x(δ(i)) = 2 ∀i∈N (27)
xe∈ {0,1} ∀e∈E (28)
whereextr(Q) andrays(Q) are the set of extreme points and rays ofQrespectively.
Notice that the problem (24)-(28) does not containf variables, but the number of constraints (25) and (26) (calledoptimality andfeasibility cuts, respectively) is very high and this suggests the use of a cutting plane
approach. In fact Benders decomposition is nothing but a cutting plane algorithm applied to the Benders reformulation. The classic version of the algorithm considers a relaxation of the Benders reformulation, called Master Problem(MP), with only a small subset of optimality and feasibility cuts. The master problem, which is a MIP itself, is solved. If the resulting DS is unbounded, then a feasibility cut is added. If DS is bounded, either the current solution is optimal or it is possible to identify a violated optimality cut. The process is then iterated.
4 Three Benders-based algorithms
We now describe three algorithms addressing theTSP-GLbased on Benders decomposition. In Section 4.1 we present the main motivations and formalize the common algorithmic scheme, called Benders-Branch-and- Cut (BBC). The three versions of the BBC differ on the type of generated feasibility cuts. In Section 4.2, we describe a basic version of the algorithm (BBC1) based on the classical Benders reformulation (24)-(28) and prove some mathematical properties. In Section 4.3, we underline the common points betweenTSP-GLand S-TSP and describe two algorithms that take advantage of such similarities (BBC2 and BBC3).
4.1 General algorithmic approach and motivations
The classic version of the Benders decomposition (CBD) requires the solution of a MIP at every master iteration. The drawback is that the complexity of such MIPs rapidly increases with the number of added constraints, which usually spoil the structure of the original problem, and convergence to optimality soon becomes impractical, except for small instances. For this reason, research focused on obtaining Benders cuts in alternative ways so as to reach a better convergence. One of the most popular techniques was proposed by McDaniel and Devine (1977). Starting from the observation that the feasible region of a dual Benders subproblem does not depend on the particular master problem solution, the authors deduce that any extreme point or ray of a dual subproblem, independently on how it was obtained, can be used to generate valid Benders cuts. Such a property implies, for example, that the solution obtained by solving the linear relaxation of the MP can be used to obtain valid Benders cuts, with the advantage of being much easier to solve than the corresponding integer version. McDaniel and Devine (1977) proposed a modification of the Benders decomposition where the initial cuts were obtained by applying the Benders scheme to the linear relaxation of the MP. When certain conditions apply, the authors propose to switch to CBD. Let us call MBD this type of solution approach. MBD has been successfully applied in several works, as for example in Cordeau et al. (2000, 2001); Rei et al. (2009).
Observe that stopping the continuous phase of the MBD when no Benders cuts can be found, corresponds to solving the linear relaxation of the initial formulation via Benders decomposition. It is actually possible to further modify MBD to progressively recover integrality by embedding the linear relaxation of the MP in a branch-and-bound scheme, instead of switching to the CBD. We name such a solution scheme Benders- Branch-and-Cut algorithm (BBC). BBC has been adopted in several works, as for example in Ljubi´c et al.
(2012); Rodr´ıguez-Mart´ın and Salazar-Gonz´alez (2008); Sridhar and Park (2000). In Naoum-Sawaya and Elhedhli (2012), BBC showed to be up to seven time faster than CBD with Pareto-optimal cuts (for details on Pareto-optimal cuts, see Magnanti and Wong 1981) when applied to capacitated facility location problem.
Our preliminary attempts to solveTSP-GLby CBD and MBD could only solve very small instances ofTSP- GL, even if a certain number of additional algorithmic improvements were implemented (Pareto-optimal cuts, heuristic generation of Benders cuts, warm-restart, etc ).
For the above reasons, the algorithms that we present here follow the BBC scheme. In order to handle
the enumeration, the BBC algorithm makes use of a list of candidate problems to be solved and can be summarized as follows:
STEP 0 Set iteration counterµ= 0, global upper boundU B = +∞. In the initial MP integrality constraints are relaxed and only degree constraints (2) are considered. In addition, a set of optimality cuts obtained heuristically is added to the formulation. The resulting starting MP is inserted into the list of problems.
STEP 1 If problem list is empty, exit. Else, select an MP from the list. Set the current problem upper bound problemU B=U B.
STEP 2 Setµ=µ+ 1. Solve MP and let (ηµ, xµ) be its solution.
STEP 3 Execute the feasibility cut separation procedure. Ifxµ violates some feasibility cut add it to MP and go back to STEP 2. Else go to STEP 4;
STEP 4 Execute the optimality separation procedure solving the Benders subproblem. Let fµ be the corre- sponding flow. If (qfµ+cxµ < problemU B), update problemU B. If (ηµ +cxµ < nodeUB - ǫ), a violated optimality cut has been found, add it to MP and go back to STEP 2. Else, current MP optimality is reached, go to STEP 5.
STEP 5 If (ηµ+cxµ > U B−ǫ) fathom the current node. Else, If (fµ, xµ) is integer, update U B. If it is not integer, a branching procedure is performed and the new problems are added to the list. Go to STEP 1.
4.2 BBC1: The basic version
The BBC1 algorithm implements the BBC scheme and adopts feasibility cuts defined by (26). Let us denote z1 the value of the optimal solution of the linear relaxation of the MP in BBC1, and zM F the value of the linear relaxation of the multicommodity flow formulation (1)-(7). It is known that the classical Benders reformulation (24)-(28) is equivalent to the original problem in terms of lower bounds provided by their linear relaxation, hencez1=zM F.
The natural extension to the full space of any extreme ray of Qhk = {(λhk, ρhk) ∈ R|A|+|N|| (λhk, ρhk) satisfy (18), . . . ,(20)} for any given (h, k) ∈ D is a ray ofQitself (for a formal proof about sets of ray generators of cones with diagonal block structure see Padberg and Sung 1991). Moreover, recall that the Benders subproblem decomposes into smaller independent subproblems, one per commodity.
Constraints (26) can be consequently substituted by the following disaggregated version:
ρhkh −ρhkk −X
e∈E
λ¯hke xe≤0 ∀(λhk, ρhk)∈extr(Qhk), ∀(h, k)∈D. (29)
The following proposition provides a result regarding the strength of these constraints.
Proposition 4.1. Let us denotezCI the value of the optimal linear relaxation of (24),(25),(27),(28)and the following Cutset Inequalities:
x(δ(S))≥1 ∀S⊂V. (30)
Then,z1=zCI
Proof. Proof. We prove the proposition by showing that inequalities (29) are equivalent to (30). To this purpose, we recall that Padberg and Sung (1991) provide a characterization of the extreme rays of the node- arc cone of a directed graph, i.e., the extreme rays ofQhk. In particular, they prove that the extreme rays of Qhk are positive multiple of vectors (λhk, ρhk) given by (i), (ii), (iii) where
(i) ρhki = 0 ∀i∈N, λhkij =
1 f or exactly one i and one j ∈N 0 otherwise
(ii) ρhki =
1 f or all i∈S
0 otherwise , λhkij =
1 f or all i∈N\S, j∈S 0 otherwise
(iii) ρhki =
−1 f or all i∈S
0 otherwise , λhkij =
1 f or all i∈S, j∈N\S 0 otherwise
andS ⊆N,1≤ |S| ≤n−1. The substitution of (i) in (29) gives the redundant inequalityxe≥0, ∀e∈E.
Let us consider now extreme rays of the form (ii) and the case h ∈ S, k ∈ N \S. Substituting in (29) one obtains the desired inequality. If bothhand kbelong toS or toN\S we obtain redundant equations.
The case (iii) is similar to (ii). Because we assumed that the demand matrix D induces a single connected component, the proposition is proved.
From the algorithmic point of view, BBC1 separates inequalities (29) by solving a minimum cost flow algorithm for each commodity, using a network simplex algorithm, for example. As soon as a violated inequality is found, the separation algorithm is stopped and the corresponding cut is added.
4.3 BBC2 and BBC3: Exploiting the Hamiltonian circuit structure
The formulation (1)-(7) does not explicitly exploit the particular Hamiltonian circuit structure of the design part ofTSP-GL. An important property that plays a central role for BBC2 and BBC3 is pointed out by the following proposition.
Proposition 4.2. Every valid inequality for the S-TSP is also valid for the TSP-GL.
Proof. Proof. LetPdenote the polyhedron defined by constraints (2)–(7), and letPS−T SP be the polyhedron of the Hamiltonian circuits defined on the same graphG. Assuming that the demand matrixDis dense enough to make subtours infeasible, calling P rojx(P) = {x∈ {0,1}|E| | ∃ f ∈ R|A||D|, (x, f) ∈P} we have that P rojx(P)⊆PS−T SP. Moreover, given anyx∈PS−T SP it is always possible to find a flow f ∈ R|A|2 such that the corresponding (x, f) belongs toP. As a consequence,PS−T SP ≡P rojx(P) and the fact that every valid inequality for the S-TSP is also valid for theTSP-GLfollows immediately.
There has been an extensive research effort to study the S-TSP polyhedron and a number of families of facet defining inequalities have been found. The above result suggests to exploit such knowledge about the S-TSP polyhedron to possibly strengthen the lower bound given by the linear relaxation of the MP. To this purpose, consider the well-known family of S-TSP facet defining inequalities called Subtour Elimination Constraints (SEC):
x(δ(S))≥2 ∀S⊂V, (31)
and let us callPM F the polyhedron of the linear relaxation (1)-(7), andPSECthe polyhedron defined byPM F
with the addition of inequalities (31). Obviously,PSEC⊆PM F, however the inclusion is tight, as shown by the following proposition.
Proposition 4.3. PSEC is strictly included inPM F
Proof. Proof. Let P rojx(PM F) = {x ∈ {0,1}|E| | ∃ f ∈ R|A||D|, (x, f) ∈ PM F} and similarly P rojx(PSEC) = {x ∈ {0,1}|E| | ∃ f ∈ R|A||D|, (x, f) ∈ PSEC} We prove the proposition by show- ing that P rojx(PSEC) is strictly included in P rojx(PM F). Recall that Benders reformulation allows us to write P rojx(PM F) = {x ∈ {0,1}|E| | (26) and (27) are satisfied} and in Proposition 4.1 we saw that this is equivalent to P rojx(PM F) = {x ∈ {0,1}|E| | (30) and (27) are satisfied}. By definition, P rojx(PSEC) ={x∈ {0,1}|E| |(26),(27) and (31) are satisfied}. The fact that inequalities (30) are clearly implied by (31) together with the observation that it is trivial to find a solution belonging toP rojx(PM F) and not toP rojx(PSEC), proves thatP rojx(PSEC) is strictly included inP rojx(PLP).
Proposition 4.3 implies that inequalities (31) can be exploited to improve the lower bound provided by BBC1.
Let us denote byPF AC the polyhedron defined by including an additional set of families of S-TSP facet defining inequalities intoPSEC. Because the added inequalities are facet defining for the S-TSP and because in Proposition 4.2 we saw that the projection on thex-space ofPM F is the S-TSP polyhedron, the following proposition easily follows:
Proposition 4.4. PF AC is strictly included inPSEC.
Therefore we may conclude that the following hierarchy of bounds holds true:
Corollary 4.1. Denote zSEC and zF AC the optimal value of minimizing the objective function (1) over polyhedraPSEC andPF AC, respectively. Then, z1=zM F ≤zSEC≤zF AC.
However, note that the improvement of zM F by adding S-TSP cuts to the original formulation (1)-(7) is far from being practical. In fact, as we will point out in Section 5.2, solving the linear relaxation of the original formulation is extremely time consuming and solution techniques requiring the iterated solution of such a problem cannot be efficient. In this sense, Benders reformulation is a key element of the success of our approach because it projects out the flow variables and makes it possible, in a BBC framework, to exploit cutting plane generation and in particular the Hamiltonian structure of the network design in theTSP-GL. BBC2 and BBC3 implement the general BBC algorithm and differ in the families of S-TSP cuts generated.
In the following, we briefly list inequalities and the adopted separation algorithms.
4.3.1 BBC2.
BBC2 is similar to BBC1, but adopts inequalities (31) as feasibility cuts instead of (26). Denotingz2 the optimal value of the linear relaxation provided by algorithm BBC2, it is clear thatz2=zSEC.
For the separation of inequalities (31), BBC2 adopts the exact method described in Padberg and Rinaldi (1990).
4.3.2 BBC3.
In addition to constraints (31) introduced by BBC2, BBC3 considers a variety of additional families of S-TSP facet defining inequalities as a subset of comb inequalities and local cuts. Let us denotez3 the optimal value of the linear relaxation provided by algorithm BBC3. From Corollary 4.1, we can clearly deduce thatz3≥z2. Recall that acombC is a subset ofN and is specified by ahandleH ⊆N and an odd number of disjoint teeth S1, . . . , Sk ⊆N such that each tooth has at least one node inH and at least one node not in H. For every combC, the corresponding inequality is:
x(δ(H)) +
k
X
j=1
x(δ(Tj))≥2k+ 1. (32)
It is not known whether the general form of comb inequalities admits a polynomial-time algorithm. How- ever, polynomial algorithms or fast heuristics for specific comb inequalities have been proposed. BBC3 imple- ments both a heuristic and an exact algorithm described in Padberg and Rao (1982) forblossoms, i.e., combs whose teeth have only two nodes. BBC3 also implements a heuristic separation of comb inequalities starting fromblocks, i.e., maximal two-connected components of the residual graph induced by the relaxed solution.
Comb inequalities from blocks are separated following the algorithm in Naddef and Thienel (2002). Finally, BBC3 implements separation procedures for local cuts. Such cuts do not follow a predefined template (as subtour elimination or comb inequalities), but are obtained by suitably shrinking nodes of the residual graph induced by a fractional solution and mapping violated inequalities back in the original graph representation.
Details on local cuts can be found in Applegate et al. (2007), Chapter 11.
5 Computational experience
We perform three experimental campaigns with the scope of 1) comparing lower bounds and their com- putational efficiency, 2) selecting the overall best algorithm, 3) testing the selected algorithm on suitable benchmark instances. More specifically, the first campaign aims at choosing which algorithm among BBC1, BBC2, and BBC3 gives the best performance in terms of root bounds and computing times. For complete- ness, we also also compare BBC algorithms with a state-of-the-art LP solver applied to (1)-(7). Since there is computational evidence of the superiority of BBC3, in the second campaign we test several possible imple- mentation options applied to BBC3 with the scope of identifying a unique algorithmic setting that averagely outperforms the others. We call the resulting algorithm BBC3+. In the third experimental campaign, we test BBC3+ on a new set of benchmark instances suitably derived from the well-known TSPLIB library.
Results show that benchmark instances with up to 42 nodes are solved to optimality, and, in the worst case, a feasible solution within 8.2% of the optimal value is obtained on a 101-nodes instance.
Given the different scopes and natures of the three experimental campaign, a suitable set of instances has been chosen for each campaign. However, construction criteria are mostly the same for all sets, and inspired by two main principles: 1) to be as general as possible 2) be representative of our main application context, i.e., the design of semi-flexible transit lines.
5.1 Instance sets
We built two instance sets each with a specific purpose: TSPGL1 to compare algorithms and algorithmic options. To guarantee statistical robustness, TSPGL1 includes a wide variety of problem types, and for each type, several randomly generated instanced are considered. TSPGL2 provides the first benchmark instance
set for theTSP-GL. For this, TSPGL2 builds upon the widely used benchmark TSPLIB and includes one instance only for each problem type. For both instance sets, the problem dimension range reflects typical situations found in our reference application.
Each instance in TSPGL1 is characterized by a set of nodes and a demand matrix D. The nodes are uniformly generated in a square of side length 100 and the graph is complete. cij is given by the Euclidean distance between nodesiandj, whilece is set equal to twice the distance between the two endpoints. This choice comes from the reference application of the TSP-GL, i.e., the design of semi-flexible transit lines, (see Errico et al. 2011b, for details), where two vehicles are considered to travel the transit line in opposite directions. The routing costsqijhk on the arcs, that model passenger travel times, are set to cijP dhk
(s,t)∈Ddst, where P dhk
(s,t)∈Ddst is the contribution of demand (h, k). Therefore P
(i,j)∈A
P
(h,k)∈Dqhkij fijhk measures the average passenger travel time.
As for demand matrices, we considered two different classes, reflecting the typical application context in which a few nodes, calledpoles, concentrate most of the transit demand. In particular, we considered the following scenarios:
• C: Complete and Polarized: The demand matrix is complete and eachdhkfollows a uniform distribution on a given interval. However, a few poles are randomly picked among the nodes and approximately half of the total demand volume has originand destination among poles.
• S: Sparse and Polarized: As in C, but the demand matrix is sparse and the number of destinations associated with a given node is limited to 4 and chosen randomly.
As specified in Errico et al. (2011b) each of the above demand matrix classes corresponds to a different approach in the design of a semi-flexible transit line, where C and S account for more or less aggregated data, respectively.
Observe that, for completeness, we initially considered also the following two set of demand matrices:
• Complete and Uniform: Demand matrix is complete anddhkare generated uniformly on a given interval.
• Sparse and Uniform: As above, but the demand matrix is sparse as the number of destinations associated with a given node is limited to 4 and chosen randomly.
The results obtained on these instances were not qualitatively different form the corresponding polarized ones.
TSPGL1 considers networks with the following number of nodes: from 15 to 40 included, by steps of 5, and 50, 60, 80, and 100. We also considered three possible values ofαin the objective function: 0.25 (emphasis on design), 0.75 (emphasis on latency), 0.5 (fairness). Finally, for each network dimension, demand setting, and objective emphasis, we generated 10 instances considering different random seeds for the generation of random data. As a result, TSPGL1 is composed of 720 different instances.
The building process for TSPGL2 is quite similar to TSPGL1, with a few important differences. First of all, the networks are deduced from the TSPLIB library instead of being generated randomly. We selected all the TSPLIB instances of the S-TSP with at most 101 cities. As in the case of TSPGL1, we considered two demand scenarios for each network, C and S. Differently from TSPGL1, TSPGL2 only considersα= 0.5 in the objective function and one only representative instance is considered for each network/demand/emphasis combination. The resulting 48 instances are available online (TSPGL2 2012).
5.2 Comparing lower bounds and efficiency
In this section, we compare root lower bounds and efficiency provided by algorithms BBC1, BBC2 and BBC3.
For completeness, we considered also the solution of the linear relaxation of the original formulation (1)-(7) by a commercial solver, more specifically, we used the dual simplex solver in CPLEX 12.3.0.0, which gave better performances with respect to other algorithms. In the following, MF will denote these results.
We implemented the BBC algorithm in the C++ programming language. The master problem was solved by the MIP solver available in CPLEX 12.3.0.0 with Concert Technology. The dual simplex was used as linear relaxation solver. Separation procedures have been implemented as instantiations of several user- cutcallback available in Concert Technology. In particular, for BBC1 the Benders subproblem has been solved via Network Simplex Optimizer implemented in CPLEX 12.0.0.3. S-TSP separation procedures in BBC2 and BBC3, i.e., subtour elimination, heuristic and exact blossom, combs from blocks, and local cuts have all been implemented by suitably importing in our codes separation procedures of the well-known S-TSP solver Concorde, version 03-12-19, coded in C programming language (Applegate et al. 2003). Details on the latter set of separation procedures can be found in Applegate et al. (2007). Experiments were executed on a Dual-Core AMD Opteron(tm) Processor 2218, with 4G DDR2 RAM memory on SunOS 5.10.
Instance GAP %Closed GAP Times (s)
dimension D. type α BBC1 BBC2 BBC3 CPLEX BBC1 BBC2 BBC3
15 C 0.25 3.4 73.6 78.3 5.4 0.9 0.9 0.9
20 C 0.25 3.2 82.3 90.6 40.1 5.0 3.3 4.5
25 C 0.25 3.3 69.1 85.3 265.9 17.1 9.6 12.9
30 C 0.25 3.5 62.8 80.3 1309.0 46.2 28.4 44.1
35 C 0.25 3.5 68.2 89.0 2872.3 75.8 50.5 57.4
15 C 0.5 2.8 66.8 69.1 8.4 2.4 1.8 2.4
20 C 0.5 3.5 52.5 61.1 69.4 13.4 7.9 11.7
25 C 0.5 4.3 45.3 57.8 486.9 39.7 20.2 36.3
30 C 0.5 4.9 33.7 50.4 1827.6 115.1 69.4 101.5
35 C 0.5 4.4 41.7 62.7 8779.2 277.8 157.8 178.7
15 C 0.75 3.4 25.9 29.6 16.2 8.5 5.4 11.4
20 C 0.75 6.6 20.9 26.0 98.7 32.3 26.5 49.4
25 C 0.75 7.8 15.4 23.3 862.6 109.4 77.7 109.1
30 C 0.75 9.2 13.2 19.7 4749.9 340.2 245.6 301.8
35 C 0.75 8.0 17.2 23.8 17495.8 742.8 494.9 470.2
15 S 0.25 3.4 74.4 77.2 5.3 0.6 0.5 0.5
20 S 0.25 3.2 80.0 89.9 22.1 2.7 1.8 2.5
25 S 0.25 3.4 67.9 84.0 149.2 7.7 4.3 6.6
30 S 0.25 3.5 62.7 79.8 552.4 22.2 12.7 18.9
35 S 0.25 3.5 67.9 88.0 1318.1 30.9 23.8 24.3
15 S 0.5 2.9 67.4 65.9 6.8 1.6 1.1 1.6
20 S 0.5 3.7 53.5 61.7 38.7 6.5 4.3 6.6
25 S 0.5 4.4 44.2 56.2 203.3 19.5 10.2 18.4
30 S 0.5 5.0 34.6 50.6 738.4 48.4 31.4 48.9
35 S 0.5 4.6 41.6 60.9 2339.8 90.6 54.0 77.1
15 S 0.75 3.5 28.1 31.6 11.8 5.2 3.1 7.9
20 S 0.75 6.9 23.3 28.1 50.9 16.6 14.7 33.5
25 S 0.75 8.5 16.3 22.9 276.2 51.4 33.7 70.9
30 S 0.75 9.5 13.4 19.7 1306.0 145.0 100.0 160.9
35 S 0.75 8.4 17.4 23.9 3397.3 275.7 189.5 270.3
Table 1: Comparison among CPLEX and BBC1, BBC2, and BBC3 on small instances
We report in Table 1 the numerical results about the comparison of MF with BBC1, BBC2, and BBC3 on small instances of the TSPGL1 set. Recalling that TSPGL1 includes 10 random samples for each instance type, rows in Table 1 report average results for each instance type. In the first three columns the instance type is identified by indicating number of nodes, demand type, and value ofα. The following column reports the relative percentage gap between the value of the optimal solution and the lower bound of the linear relaxation provided byzMF. Given that, as observed in Section 4.2, zM F =z1, there is no need to report
the lower bound quality of BBC1. The next two columns display the average gap percentage closed by z2
andz3with respect to zM F. The last four columns report the average computing times in seconds.
We observe that the lower bound provided by BBC1 (equivalent to MF) is the worst and has a gap ranging from about 3% to no more than 10%. This gap increases with size of the instances and is also affected by the value ofα. Whenαis higher, i.e., when the travel time component of the objective function is greater, the gap increases, while when the design component dominates, thus when the instances are closer to a TSP problem, the bound is tighter. Note also that the gap of BBC1 is not affected by the type (C/S) of the demand matrix. As for the computing times, there is a huge difference between the commercial software and BBC1. CPLEX times dramatically increase with the size and also testify that instances with high values of αare more difficult. Moreover, there is a great impact on the times due to the demand matrix type. Dense instances (C) require much higher times with respect to the sparse ones (S). This is mainly due to the number of variables of the model. Such a difference in computing times is mitigated using BBC1 instead of CPLEX.
Besides requiring much smaller times, BBC1 reveals also to be more stable with respect all parameters.
Considering the other two bounds, we may observe a clear improvement in terms of gap and computing time. The gap closed by BBC2 with respect to BBC1 ranges in average from 80% for the easy instances to 13% for the more difficult ones. Notice also that the time of BBC2 is shorter that that of BBC1. BBC3 reaches better bounds closing a larger amount of gap and has computing times comparable with BBC1 and sometimes smaller for big instances, thus showing a better trend with respect to the instance size.
It is interesting to notice how the TSP-GL behaves differently than the general FNDP family. In the latter problems in fact, the more important the flow component with respect to the design component is, the easiest is the problem. This is normally explained noticing that the linear part of the problem (the flows) becomes predominant with respect to the non linear one (the design). For the TSP-GL the situation is different because the Hamiltonian structure of the network topology seems to emphasize the non-linearity of the problem when flow costs are higher. From the computational view point, we observed for BBC algorithms that the number of optimality cuts generated is much higher for high values ofα. This suggests that the optimal solution of the master problem is located towards the interior of the feasible region of the projection in thex-subspace of theTSP-GLpolyhedron. This would also explain why optimal solutions of the linear relaxation are more fractional, with consequently worse lower bounds, and why S-TSP facet defining inequalities are less helpful for highα.
We performed a second experimental test where we compared BBC1, BBC2 and BBC2 on instances of TSPGL1 with a higher number of nodes, and limiting the computing time to 5 hours. In this second test, we did not compare with MF because the commercial solver usually failed due to memory limits. Also, differently from the previous experimentation, we do not know the values of the optimal solutions of the integer problems. We consequently compared the relative improvements of lower bound values instead of comparing the relative gaps with the optimum. Results are reported in Table 2.
The meaning of the columns is as follows: the first three columns identify the instance type by specifying the number of nodes, the demand type and the value ofα, respectively. The next column reports the average value of BBC1, on the 10 considered random seeds, while the following two columns report the average relative improvement of BBC2 and BBC3 over BBC1, respectively, computed as 100(zi−z1)/z1, i= 2,3.
The next three columns represent the average computing times, while the last three columns report the number instances for which the algorithm stops due to time limit reasons.
Table 2 confirms the trends observed in Table 1. Perhaps the most important fact underlined by Table 2 is that BBC3 consistently outperforms BBC1 and BBC2 in terms of lower bounds. Its computing times are comparable with BBC2, and are even better on harder instances, as it can be deduced by comparing how many times BBC3 hits the time limit with respect to BBC1 and BBC2.
Instance GAP GAP %Impr. Times (sec) #Time limit
dimension D. type α BBC1 BBC2 BBC3 BBC1 BBC2 BBC3 BBC1 BBC2 BBC3
40 C 0.25 761.8 2.6 3.4 166 95 132 0 0 0
50 C 0.25 848.1 3.3 4.1 599 351 469 0 0 0
60 C 0.25 916.6 3.1 4.0 1354 1079 1177 0 0 0
80 C 0.25 1040.8 3.1 4.0 7266 4951 6417 0 0 0
100 C 0.25 1148.0 3.4 4.3 17937 16392 16219 9 7 6
40 C 0.5 538.2 2.3 2.9 509 393 353 0 0 0
50 C 0.5 597.0 2.9 3.6 2060 1505 1297 0 0 0
60 C 0.5 640.7 2.6 3.4 6466 4105 3340 1 0 0
80 C 0.5 726.7 2.5 3.3 17604 15836 15906 8 6 4
100 C 0.5 793.7 3.3 4.1 18000 18000 18000 10 10 10
40 C 0.75 309.3 1.7 2.1 1371 1088 1062 0 0 0
50 C 0.75 339.9 2.2 2.6 5378 4805 4237 0 0 0
60 C 0.75 358.9 1.9 2.4 15634 11760 10576 4 0 0
80 C 0.75 401.8 2.5 3.2 18000 18000 18000 10 10 10
100 C 0.75 414.0 5.8 7.0 18000 18000 18000 10 10 10
40 S 0.25 761.3 2.7 3.5 1860 38 53 1 0 0
50 S 0.25 847.6 3.3 4.1 203 116 164 0 0 0
60 S 0.25 915.7 3.1 4.0 492 345 404 0 0 0
80 S 0.25 1040.6 3.1 4.0 2477 1398 1632 0 0 0
100 S 0.25 1147.4 3.4 4.3 8105 5896 6916 0 0 0
40 S 0.5 537.3 2.4 3.0 1975 144 154 1 0 0
50 S 0.5 596.0 3.0 3.6 637 484 484 0 0 0
60 S 0.5 639.2 2.8 3.5 1525 1154 1269 0 0 0
80 S 0.5 726.3 2.5 3.4 7561 4886 5026 0 0 0
100 S 0.5 796.1 3.0 3.8 17932 17490 15872 9 9 5
40 S 0.75 308.0 1.8 2.2 494 375 485 0 0 0
50 S 0.75 338.3 2.3 2.7 1681 1448 1526 0 0 0
60 S 0.75 356.9 2.1 2.6 6429 3905 3849 1 0 0
80 S 0.75 405.1 1.8 2.5 17988 16162 16057 9 4 4
100 S 0.75 433.3 3.2 3.9 18000 18000 18000 10 10 10
Table 2: Comparison among BBC1, BBC2 and BBC3 on bigger INST1 instances. Time limit: 5h
